MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  carden2b Structured version   Visualization version   GIF version

Theorem carden2b 10005
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 10004 are meant to replace carden 10589 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 10003 . . . . 5 ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2 ennum 9985 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
32biimpa 476 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4 cardid2 9991 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
53, 4syl 17 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6 ensym 9042 . . . . . . 7 (𝐴𝐵𝐵𝐴)
76adantr 480 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵𝐴)
8 entr 9045 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵𝐴) → (card‘𝐵) ≈ 𝐴)
95, 7, 8syl2anc 584 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
101, 9nsyl3 138 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11 cardon 9982 . . . . 5 (card‘𝐴) ∈ On
12 cardon 9982 . . . . 5 (card‘𝐵) ∈ On
13 ontri1 6420 . . . . 5 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
1411, 12, 13mp2an 692 . . . 4 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
1510, 14sylibr 234 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16 cardne 10003 . . . . 5 ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17 cardid2 9991 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18 id 22 . . . . . 6 (𝐴𝐵𝐴𝐵)
19 entr 9045 . . . . . 6 (((card‘𝐴) ≈ 𝐴𝐴𝐵) → (card‘𝐴) ≈ 𝐵)
2017, 18, 19syl2anr 597 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
2116, 20nsyl3 138 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22 ontri1 6420 . . . . 5 (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
2312, 11, 22mp2an 692 . . . 4 ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
2421, 23sylibr 234 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
2515, 24eqssd 4013 . 2 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26 ndmfv 6942 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
2726adantl 481 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
282notbid 318 . . . . 5 (𝐴𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
2928biimpa 476 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30 ndmfv 6942 . . . 4 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3129, 30syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
3227, 31eqtr4d 2778 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
3325, 32pm2.61dan 813 1 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963  c0 4339   class class class wbr 5148  dom cdm 5689  Oncon0 6386  cfv 6563  cen 8981  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-card 9977
This theorem is referenced by:  card1  10006  carddom2  10015  cardennn  10021  cardsucinf  10022  pm54.43lem  10038  nnadju  10236  nnadjuALT  10237  ficardun  10239  ackbij1lem5  10261  ackbij1lem8  10264  ackbij1lem9  10265  ackbij2lem2  10277  carden  10589  r1tskina  10820  cardfz  14008
  Copyright terms: Public domain W3C validator